Subespaços complementados e não complementados em espaços de polinômios homogêneos

Abstract

The main objective of this work is to study complemented and uncomplemented subspaces of spaces of homogeneous polynomials between Banach spaces. Initially, several classical cases of complemented subspaces are examined in detail. Next, the focus shifts to the subspaces formed by homogeneous polynomials that are compact, weakly compact, and weakly continuous on bounded sets, as well as situations in which these subspaces are not complemented. This study is based on a significant result by I. Ghenciu concerning uncomplemented subspaces of spaces of linear operators, whose proof is developed in detail. Finally, we present the study of its generalized version for polynomials, due to S. Pérez. The polynomial theorems are proved thoroughly, and numerous applications to specific cases are provided. Through this approach, various conditions under which the aforementioned polynomial subspaces are uncomplemented are established.